The category of Nori motives $NMM$ is defined by tools of graph category. I think one can similarly define a category $EM$ as the graph category of $P(k)$ (the graph of projective smooth k-varieties) where the fiber functor is given by Hodge realization, is this category, after inverting $1(1)$, equivalent to the category $M(k)$ of pure motives defined by Grothendieck? (The universality of $EM$ gives the map $EM\to M(k)_{eff}$.)

Besides, has anyone proved that the category of pure motives is fully embedded into $NMM$?

pure motives- There's no such thing... ;-) $\endgroup$ – Lucian Oct 18 '14 at 7:58